MHB Demand for a certain Commodity

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The demand for a certain commodity is represented by the function D(x) = 1000e^(-0.03x), and consumer expenditure is calculated as E(x) = xD(x). For part (a), the rate of change of consumer expenditure at a price of $160 is found to be approximately -31.27. To determine when consumer expenditure stops increasing, the derivative E'(x) must equal zero, indicating a transition from positive to negative values. The discussion also raises questions about the value of "e" in the context of the exponential function and the conditions under which E' becomes positive, signaling an increase in consumer expenditure.
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The demand for a certain commodity is

D(x)  =  1000e−.03x

units per month when the market price is x dollars per unit.

(a) At what rate is the consumer expenditure E(x) = xD(x) changing with respect to price x when the price is equal to $160 dollars?
(b) At what price does consumer expenditure stop increasing and begin to decrease?
(c) At what price does the rate of consumer expenditure begin to increase?

I am not sure but I got -31.27 for a) but i really have no idea how to go about this question.
 
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confusedbycalc said:
The demand for a certain commodity is

D(x)  =  1000e−.03x
What is "e" here? (I suspect it is not 2.718...

units per month when the market price is x dollars per unit.

(a) At what rate is the consumer expenditure E(x) = xD(x) changing with respect to price x when the price is equal to $160 dollars?
With D(x)= 1000e- 0.3x, E(x)= 1000ex- 0.3x^2. The rate of change of that is the derivative E'(x)= 1000e- 0.6x.

(b) At what price does consumer expenditure stop increasing and begin to decrease?
As long as E' is positive the consumer expenditure is increasing. It is decreasing when E' is negative. To change from positive to negative, E' has to become 0.

(c) At what price does the rate of consumer expenditure begin to increase?
when is E' positive?

I am not sure but I got -31.27 for a) but i really have no idea how to go about this question.
 

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